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Explicit triangular decoupling of the separated Lichnerowicz tensor wave equation on Schwarzschild into scalar Regge-Wheeler equations. (English) Zbl 1490.35488

Summary: We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse triangular form, where the diagonal blocks consist of spin-\(s\) scalar Regge-Wheeler equations (for spins \(s=0,1,2)\). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
68W30 Symbolic computation and algebraic computation
83C57 Black holes
83C10 Equations of motion in general relativity and gravitational theory
83C35 Gravitational waves
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C22 Einstein-Maxwell equations

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